Perfect Square Trinomials worksheet with algebra problems to factor.
Worksheet titled "Perfect Square Trinomials" with instructions to factor each trinomial, featuring nine algebraic expressions arranged in a grid, decorative heart icons, and a watermark "WORKSHEET ZONE".
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Step-by-step solution for: Perfect Square Trinomials Math Worksheet Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Perfect Square Trinomials Math Worksheet Worksheet
To solve the problem of factoring each perfect square trinomial, we need to recognize that a perfect square trinomial is of the form:
\[
a^2 + 2ab + b^2 = (a + b)^2
\]
or
\[
a^2 - 2ab + b^2 = (a - b)^2
\]
We will factor each trinomial step by step.
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 25 \) so \( b = 5 \)
- The middle term is \( 2ab = 2(x)(5) = 10x \), which matches.
- Factor:
\[
x^2 + 10x + 25 = (x + 5)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 16 \) so \( b = 4 \)
- The middle term is \( -2ab = -2(x)(4) = -8x \), which matches.
- Factor:
\[
x^2 - 8x + 16 = (x - 4)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 9 \) so \( b = 3 \)
- The middle term is \( 2ab = 2(x)(3) = 6x \), which matches.
- Factor:
\[
x^2 + 6x + 9 = (x + 3)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 4 \) so \( b = 2 \)
- The middle term is \( -2ab = -2(x)(2) = -4x \), which matches.
- Factor:
\[
x^2 - 4x + 4 = (x - 2)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 81 \) so \( b = 9 \)
- The middle term is \( 2ab = 2(x)(9) = 18x \), which matches.
- Factor:
\[
x^2 + 18x + 81 = (x + 9)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 100 \) so \( b = 10 \)
- The middle term is \( -2ab = -2(x)(10) = -20x \), which matches.
- Factor:
\[
x^2 - 20x + 100 = (x - 10)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = 4x^2 \) so \( a = 2x \)
- \( b^2 = 9 \) so \( b = 3 \)
- The middle term is \( 2ab = 2(2x)(3) = 12x \), which matches.
- Factor:
\[
4x^2 + 12x + 9 = (2x + 3)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = 9x^2 \) so \( a = 3x \)
- \( b^2 = 16 \) so \( b = 4 \)
- The middle term is \( -2ab = -2(3x)(4) = -24x \), which matches.
- Factor:
\[
9x^2 - 24x + 16 = (3x - 4)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = 9x^2 \) so \( a = 3x \)
- \( b^2 = 4 \) so \( b = 2 \)
- The middle term is \( -2ab = -2(3x)(2) = -12x \), which matches.
- Factor:
\[
9x^2 - 12x + 4 = (3x - 2)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = 25x^2 \) so \( a = 5x \)
- \( b^2 = 1 \) so \( b = 1 \)
- The middle term is \( 2ab = 2(5x)(1) = 10x \), which matches.
- Factor:
\[
25x^2 + 10x + 1 = (5x + 1)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = 16x^2 \) so \( a = 4x \)
- \( b^2 = 81 \) so \( b = 9 \)
- The middle term is \( -2ab = -2(4x)(9) = -72x \), which matches.
- Factor:
\[
16x^2 - 72x + 81 = (4x - 9)^2
\]
- Identify \( a \) and \( b \):
- \( a^2 = 4x^2 \) so \( a = 2x \)
- \( b^2 = 49 \) so \( b = 7 \)
- The middle term is \( 2ab = 2(2x)(7) = 28x \), which matches.
- Factor:
\[
4x^2 + 28x + 49 = (2x + 7)^2
\]
\[
\boxed{
\begin{aligned}
&x^2 + 10x + 25 = (x + 5)^2 \\
&x^2 - 8x + 16 = (x - 4)^2 \\
&x^2 + 6x + 9 = (x + 3)^2 \\
&x^2 - 4x + 4 = (x - 2)^2 \\
&x^2 + 18x + 81 = (x + 9)^2 \\
&x^2 - 20x + 100 = (x - 10)^2 \\
&4x^2 + 12x + 9 = (2x + 3)^2 \\
&9x^2 - 24x + 16 = (3x - 4)^2 \\
&9x^2 - 12x + 4 = (3x - 2)^2 \\
&25x^2 + 10x + 1 = (5x + 1)^2 \\
&16x^2 - 72x + 81 = (4x - 9)^2 \\
&4x^2 + 28x + 49 = (2x + 7)^2
\end{aligned}
}
\]
\[
a^2 + 2ab + b^2 = (a + b)^2
\]
or
\[
a^2 - 2ab + b^2 = (a - b)^2
\]
We will factor each trinomial step by step.
1. \( x^2 + 10x + 25 \)
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 25 \) so \( b = 5 \)
- The middle term is \( 2ab = 2(x)(5) = 10x \), which matches.
- Factor:
\[
x^2 + 10x + 25 = (x + 5)^2
\]
2. \( x^2 - 8x + 16 \)
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 16 \) so \( b = 4 \)
- The middle term is \( -2ab = -2(x)(4) = -8x \), which matches.
- Factor:
\[
x^2 - 8x + 16 = (x - 4)^2
\]
3. \( x^2 + 6x + 9 \)
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 9 \) so \( b = 3 \)
- The middle term is \( 2ab = 2(x)(3) = 6x \), which matches.
- Factor:
\[
x^2 + 6x + 9 = (x + 3)^2
\]
4. \( x^2 - 4x + 4 \)
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 4 \) so \( b = 2 \)
- The middle term is \( -2ab = -2(x)(2) = -4x \), which matches.
- Factor:
\[
x^2 - 4x + 4 = (x - 2)^2
\]
5. \( x^2 + 18x + 81 \)
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 81 \) so \( b = 9 \)
- The middle term is \( 2ab = 2(x)(9) = 18x \), which matches.
- Factor:
\[
x^2 + 18x + 81 = (x + 9)^2
\]
6. \( x^2 - 20x + 100 \)
- Identify \( a \) and \( b \):
- \( a^2 = x^2 \) so \( a = x \)
- \( b^2 = 100 \) so \( b = 10 \)
- The middle term is \( -2ab = -2(x)(10) = -20x \), which matches.
- Factor:
\[
x^2 - 20x + 100 = (x - 10)^2
\]
7. \( 4x^2 + 12x + 9 \)
- Identify \( a \) and \( b \):
- \( a^2 = 4x^2 \) so \( a = 2x \)
- \( b^2 = 9 \) so \( b = 3 \)
- The middle term is \( 2ab = 2(2x)(3) = 12x \), which matches.
- Factor:
\[
4x^2 + 12x + 9 = (2x + 3)^2
\]
8. \( 9x^2 - 24x + 16 \)
- Identify \( a \) and \( b \):
- \( a^2 = 9x^2 \) so \( a = 3x \)
- \( b^2 = 16 \) so \( b = 4 \)
- The middle term is \( -2ab = -2(3x)(4) = -24x \), which matches.
- Factor:
\[
9x^2 - 24x + 16 = (3x - 4)^2
\]
9. \( 9x^2 - 12x + 4 \)
- Identify \( a \) and \( b \):
- \( a^2 = 9x^2 \) so \( a = 3x \)
- \( b^2 = 4 \) so \( b = 2 \)
- The middle term is \( -2ab = -2(3x)(2) = -12x \), which matches.
- Factor:
\[
9x^2 - 12x + 4 = (3x - 2)^2
\]
10. \( 25x^2 + 10x + 1 \)
- Identify \( a \) and \( b \):
- \( a^2 = 25x^2 \) so \( a = 5x \)
- \( b^2 = 1 \) so \( b = 1 \)
- The middle term is \( 2ab = 2(5x)(1) = 10x \), which matches.
- Factor:
\[
25x^2 + 10x + 1 = (5x + 1)^2
\]
11. \( 16x^2 - 72x + 81 \)
- Identify \( a \) and \( b \):
- \( a^2 = 16x^2 \) so \( a = 4x \)
- \( b^2 = 81 \) so \( b = 9 \)
- The middle term is \( -2ab = -2(4x)(9) = -72x \), which matches.
- Factor:
\[
16x^2 - 72x + 81 = (4x - 9)^2
\]
12. \( 4x^2 + 28x + 49 \)
- Identify \( a \) and \( b \):
- \( a^2 = 4x^2 \) so \( a = 2x \)
- \( b^2 = 49 \) so \( b = 7 \)
- The middle term is \( 2ab = 2(2x)(7) = 28x \), which matches.
- Factor:
\[
4x^2 + 28x + 49 = (2x + 7)^2
\]
Final Answer:
\[
\boxed{
\begin{aligned}
&x^2 + 10x + 25 = (x + 5)^2 \\
&x^2 - 8x + 16 = (x - 4)^2 \\
&x^2 + 6x + 9 = (x + 3)^2 \\
&x^2 - 4x + 4 = (x - 2)^2 \\
&x^2 + 18x + 81 = (x + 9)^2 \\
&x^2 - 20x + 100 = (x - 10)^2 \\
&4x^2 + 12x + 9 = (2x + 3)^2 \\
&9x^2 - 24x + 16 = (3x - 4)^2 \\
&9x^2 - 12x + 4 = (3x - 2)^2 \\
&25x^2 + 10x + 1 = (5x + 1)^2 \\
&16x^2 - 72x + 81 = (4x - 9)^2 \\
&4x^2 + 28x + 49 = (2x + 7)^2
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of perfect square worksheet.
